All-loop-orders relation between Regge limits of ${\cal N}=4$ SYM and ${\cal N}=8$ supergravity four-point amplitudes

TL;DR

This paper analyzes the Regge limit of nonplanar ${ m N}=4$ SYM four-point amplitudes across multiple loops, verifying IR divergence structures and proposing an all-orders relation with ${ m N}=8$ supergravity amplitudes.

Contribution

It develops a color factor basis for the Regge limit, computes amplitude coefficients up to three loops, and conjectures an all-orders relation with ${ m N}=8$ supergravity.

Findings

Verified IR divergence consistency with dipole formula

Computed amplitude coefficients explicitly at three loops

Proposed an all-orders relation with supergravity amplitudes

Abstract

We examine in detail the structure of the Regge limit of the (nonplanar) ${\cal N}=4$ SYM four-point amplitude. We begin by developing a basis of color factors $C_{ik}$ suitable for the Regge limit of the amplitude at any loop order, and then calculate explicitly the coefficients of the amplitude in that basis through three-loop order using the Regge limit of the full amplitude previously calculated by Henn and Mistlberger. We compute these coefficients exactly at one loop, through ${\cal O} (\epsilon^2)$ at two loops, and through ${\cal O} (\epsilon^0)$ at three loops, verifying that the IR-divergent pieces are consistent with (the Regge limit of) the expected infrared divergence structure, including a contribution from the three-loop correction to the dipole formula. We also verify consistency with the IR-finite NLL and NNLL predictions of Caron-Huot et al. Finally we use these results to motivate the conjecture of an all-orders relation between one of the coefficients and the Regge limit of the ${\cal N} =8$ supergravity four-point amplitude.